The idea of using spread measures to bound thresholds comes from the fractional expectation threshold vs. threshold conjecture of Talagrand [1]} which was only recently established in a breakthrough due to Frankston, Kahn, Narayanan, and Park [2]}. We note that this result is a fractional version of the Kahn–Kalai expectation threshold vs. threshold conjecture [3]} and that the full conjecture, which implies the fractional version, was resolved in very recent work of Park and the first author [4]}. For our purposes, the crucial corollary is a connection between so-called spread measures and thresholds.
